Models need to be trained with privacy-preserving learning algorithms to prevent leakage of possibly sensitive information contained in their training data. However, canonical algorithms like differentially private stochastic gradient descent (DP-SGD) do not benefit from model scale in the same way as non-private learning. This manifests itself in the form of unappealing tradeoffs between privacy and utility (accuracy) when using DP-SGD on complex tasks. To remediate this tension, a paradigm is emerging: fine-tuning with differential privacy from a model pretrained on public (i.e., non-sensitive) training data. In this work, we identify an oversight of existing approaches for differentially private fine tuning. They do not tailor the fine-tuning approach to the specifics of learning with privacy. Our main result is to show how carefully selecting the layers being fine-tuned in the pretrained neural network allows us to establish new state-of-the-art tradeoffs between privacy and accuracy. For instance, we achieve 77.9% accuracy for $(\varepsilon, \delta)=(2, 10^{-5})$ on CIFAR-100 for a model pretrained on ImageNet. Our work calls for additional hyperparameter search to configure the differentially private fine-tuning procedure itself.

Continual learning in neural networks suffers from a phenomenon called catastrophic forgetting, in which a network quickly forgets what was learned in a previous task. The human brain, however, is able to continually learn new tasks and accumulate knowledge throughout life. Neuroscience findings suggest that continual learning success in the human brain is potentially associated with its modular structure and memory consolidation mechanisms. In this paper we propose a novel topological regularization that penalizes cycle structure in a neural network during training using principled theory from persistent homology and optimal transport. The penalty encourages the network to learn modular structure during training. The penalization is based on the closed-form expressions of the Wasserstein distance and barycenter for the topological features of a 1-skeleton representation for the network. Our topological continual learning method combines the proposed regularization with a tiny episodic memory to mitigate forgetting. We demonstrate that our method is effective in both shallow and deep network architectures for multiple image classification datasets.

Optimizing machine learning algorithms that are used to solve the objective function has been of great interest. Several approaches to optimize common algorithms, such as gradient descent and stochastic gradient descent, were explored. One of these approaches is reducing the gradient variance through adaptive sampling to solve large-scale optimization's empirical risk minimization (ERM) problems. In this paper, we will explore how starting with a small sample and then geometrically increasing it and using the solution of the previous sample ERM to compute the new ERM. This will solve ERM problems with first-order optimization algorithms of sublinear convergence but with lower computational complexity. This paper starts with theoretical proof of the approach, followed by two experiments comparing the gradient descent with the adaptive sampling of the gradient descent and ADAM with adaptive sampling ADAM on different datasets.

Driven by several successful applications such as in stochastic gradient descent or in Bayesian computation, control variates have become a major tool for Monte Carlo integration. However, standard methods do not allow the distribution of the particles to evolve during the algorithm, as is the case in sequential simulation methods. Within the standard adaptive importance sampling framework, a simple weighted least squares approach is proposed to improve the procedure with control variates. The procedure takes the form of a quadrature rule with adapted quadrature weights to reflect the information brought in by the control variates. The quadrature points and weights do not depend on the integrand, a computational advantage in case of multiple integrands. Moreover, the target density needs to be known only up to a multiplicative constant. Our main result is a non-asymptotic bound on the probabilistic error of the procedure. The bound proves that for improving the estimate's accuracy, the benefits from adaptive importance sampling and control variates can be combined. The good behavior of the method is illustrated empirically on synthetic examples and real-world data for Bayesian linear regression.

Bayesian optimization (BO) is a popular method for efficiently inferring optima of an expensive black-box function via a sequence of queries. Existing information-theoretic BO procedures aim to make queries that most reduce the uncertainty about optima, where the uncertainty is captured by Shannon entropy. However, an optimal measure of uncertainty would, ideally, factor in how we intend to use the inferred quantity in some downstream procedure. In this paper, we instead consider a generalization of Shannon entropy from work in statistical decision theory (DeGroot 1962, Rao 1984), which contains a broad class of uncertainty measures parameterized by a problem-specific loss function corresponding to a downstream task. We first show that special cases of this entropy lead to popular acquisition functions used in BO procedures such as knowledge gradient, expected improvement, and entropy search. We then show how alternative choices for the loss yield a flexible family of acquisition functions that can be customized for use in novel optimization settings. Additionally, we develop gradient-based methods to efficiently optimize our proposed family of acquisition functions, and demonstrate strong empirical performance on a diverse set of sequential decision making tasks, including variants of top-$k$ optimization, multi-level set estimation, and sequence search.

Adversarial training (AT) is a widely recognized defense mechanism to gain the robustness of deep neural networks against adversarial attacks. It is built on min-max optimization (MMO), where the minimizer (i.e., defender) seeks a robust model to minimize the worst-case training loss in the presence of adversarial examples crafted by the maximizer (i.e., attacker). However, the conventional MMO method makes AT hard to scale. Thus, Fast-AT (Wong et al., 2020) and other recent algorithms attempt to simplify MMO by replacing its maximization step with the single gradient sign-based attack generation step. Although easy to implement, Fast-AT lacks theoretical guarantees, and its empirical performance is unsatisfactory due to the issue of robust catastrophic overfitting when training with strong adversaries. In this paper, we advance Fast-AT from the fresh perspective of bi-level optimization (BLO). We first show that the commonly-used Fast-AT is equivalent to using a stochastic gradient algorithm to solve a linearized BLO problem involving a sign operation. However, the discrete nature of the sign operation makes it difficult to understand the algorithm performance. Inspired by BLO, we design and analyze a new set of robust training algorithms termed Fast Bi-level AT (Fast-BAT), which effectively defends sign-based projected gradient descent (PGD) attacks without using any gradient sign method or explicit robust regularization. In practice, we show our method yields substantial robustness improvements over baselines across multiple models and datasets. Codes are available at https://github.com/OPTML-Group/Fast-BAT.

We show that, although solution trajectories of gradient-based algorithms may vary depending on the learning task, they behave almost monotonically when projected onto an appropriate orthonormal function basis. Such projection gives rise to a basis function decomposition of the solution trajectory. Theoretically, we use our proposed basis function decomposition to establish the convergence of gradient descent (GD) on several representative learning tasks. In particular, we improve the convergence of GD on symmetric matrix factorization and provide a completely new convergence result for the orthogonal symmetric tensor decomposition. Empirically, we illustrate the promise of our proposed framework on realistic deep neural networks (DNNs) across different architectures, gradient-based solvers, and datasets. Our key finding is that gradient-based algorithms monotonically learn the coefficients of a particular orthonormal function basis of DNNs defined as the eigenvectors of the conjugate kernel after training.

Continual learning aims to alleviate catastrophic forgetting when handling consecutive tasks under non-stationary distributions. Gradient-based meta-learning algorithms have shown the capability to implicitly solve the transfer-interference trade-off problem between different examples. However, they still suffer from the catastrophic forgetting problem in the setting of continual learning, since the past data of previous tasks are no longer available. In this work, we propose a novel efficient meta-learning algorithm for solving the online continual learning problem, where the regularization terms and learning rates are adapted to the Taylor approximation of the parameter's importance to mitigate forgetting. The proposed method expresses the gradient of the meta-loss in closed-form and thus avoid computing second-order derivative which is computationally inhibitable. We also use Proximal Gradient Descent to further improve computational efficiency and accuracy. Experiments on diverse benchmarks show that our method achieves better or on-par performance and much higher efficiency compared to the state-of-the-art approaches.

As more practical and scalable quantum computers emerge, much attention has been focused on realizing quantum supremacy in machine learning. Existing quantum ML methods either (1) embed a classical model into a target Hamiltonian to enable quantum optimization or (2) represent a quantum model using variational quantum circuits and apply classical gradient-based optimization. The former method leverages the power of quantum optimization but only supports simple ML models, while the latter provides flexibility in model design but relies on gradient calculation, resulting in barren plateau (i.e., gradient vanishing) and frequent classical-quantum interactions. To address the limitations of existing quantum ML methods, we introduce Quark, a gradient-free quantum learning framework that optimizes quantum ML models using quantum optimization. Quark does not rely on gradient computation and therefore avoids barren plateau and frequent classical-quantum interactions. In addition, Quark can support more general ML models than prior quantum ML methods and achieves a dataset-size-independent optimization complexity. Theoretically, we prove that Quark can outperform classical gradient-based methods by reducing model query complexity for highly non-convex problems; empirically, evaluations on the Edge Detection and Tiny-MNIST tasks show that Quark can support complex ML models and significantly reduce the number of measurements needed for discovering near-optimal weights for these tasks.

Decision trees provide a rich family of highly non-linear but efficient models, due to which they continue to be the go-to family of predictive models by practitioners across domains. But learning trees is challenging due to their discrete decision boundaries. The state-of-the-art (SOTA) techniques resort to (a) learning \textit{soft} trees thereby losing logarithmic inference time; or (b) using methods tailored to specific supervised learning settings, requiring access to labeled examples and loss function. In this work, by leveraging techniques like overparameterization and straight-through estimators, we propose a unified method that enables accurate end-to-end gradient based tree training and can be deployed in a variety of settings like offline supervised learning and online learning with bandit feedback. Using extensive validation on standard benchmarks, we demonstrate that our method provides best of both worlds, i.e., it is competitive to, and in some cases more accurate than methods designed \textit{specifically} for the supervised settings; and in bandit settings, where most existing tree learning techniques are not applicable, our models are still accurate and significantly outperform the applicable SOTA methods.